- Test each candidate function by substituting the provided x-values to see if they result in the matching f(x) values from the table.
- Verify \(f(x) = 2x + 3\) by checking it against all points: \((-7, -11)\), \((-1, 1)\), \((3, 9)\), \((4, 11)\), and \((7, 17)\).
- Observe that all other options fail for at least one data point.
- The correct function rule is \(\boxed{f(x) = 2x + 3}\).
Explanation
- Analysis of the Problem and Data Points
We are given a set of data points \((x, f(x))\) as \((-7, -11), (-1, 1), (3, 9), (4, 11), (7, 17)\) and need to identify the correct linear function \(f(x) = mx + b\) from the provided options:
- \(f(x) = 3x + 10\)
- \(f(x) = 2x + 3\)
- \(f(x) = 4x + 5\)
- \(f(x) = 3x - 10\)
To determine the correct rule, we will test each candidate function against the provided data points. If a function holds true for all points in the table, it is the correct model.
- Evaluating Candidate Function Rules
Let's test each candidate function by substituting the given \(x\)-values and checking if the resulting \(f(x)\) matches the table:
- Testing \(f(x) = 3x + 10\):
- For \(x = -7\): \(3(-7) + 10 = -21 + 10 = -11\) (Match)
- For \(x = -1\): \(3(-1) + 10 = -3 + 10 = 7
eq 1\) (No match, so this rule is incorrect)
- Testing \(f(x) = 2x + 3\):
- For \(x = -7\): \(2(-7) + 3 = -14 + 3 = -11\) (Match)
- For \(x = -1\): \(2(-1) + 3 = -2 + 3 = 1\) (Match)
- For \(x = 3\): \(2(3) + 3 = 6 + 3 = 9\) (Match)
- For \(x = 4\): \(2(4) + 3 = 8 + 3 = 11\) (Match)
- For \(x = 7\): \(2(7) + 3 = 14 + 3 = 17\) (Match)
Since all points match, this is the correct function rule.
- Testing \(f(x) = 4x + 5\):
- For \(x = -7\): \(4(-7) + 5 = -28 + 5 = -23
eq -11\) (No match)
- Testing \(f(x) = 3x - 10\):
- For \(x = -7\): \(3(-7) - 10 = -21 - 10 = -31
eq -11\) (No match)
- Conclusion
After testing each of the provided options against the given coordinate pairs, we have confirmed that the function \(f(x) = 2x + 3\) correctly predicts all the values in the table. Therefore, this function is the correct model for the data provided. The final answer is \(\boxed{f(x) = 2x + 3}\).
Examples
Linear functions are essential for modeling relationships where a constant change occurs, such as in finance or physics. For instance, if you are planning a trip, the total cost might include a base fee (the y-intercept, \(b\)) plus a cost per mile traveled (the slope, \(m\)). By collecting data points on total trip costs for various distances, you can determine the linear model \(f(x) = mx + b\) to predict the price for any future journey. This application helps individuals and businesses make informed budget decisions based on historical trends.